A053026 Maximum power of 2 arising when A000005 is applied repeatedly to n!.

1, 2, 4, 8, 16, 8, 4, 4, 4, 16, 8, 8, 8, 8, 8, 2, 8, 4, 8, 4, 4, 4, 4, 4, 4, 4, 16, 16, 4, 16, 8, 16, 4, 4, 16, 4, 4, 2, 4, 16, 8, 16, 16, 4, 8, 8, 4, 8, 8, 16, 8, 8, 32, 32, 4, 32, 4, 4, 8, 4, 2, 32, 2, 8, 4, 8, 4, 8, 8, 8, 8, 2, 8, 8, 8, 32, 32, 8, 4, 8, 8, 4, 8, 8, 8, 8, 8, 32, 8, 8, 2, 4, 2, 4, 8

Offset: 1

Author

Labos Elemer, Feb 24 2000

Keywords

• nonn

Comments

Unlike the iteration of Euler phi (A000010) or cototient (A051953) functions, here the emerging powers of 2 are not accumulated at the terminal phase of iteration sequence. Non-2-powers can be intercalated.

Examples

			For n = 53, the iterations are {53!, 16174080000, 840, 32, 6, 4, 3, 2}, so a(53) = 32.
For n = 130, the iterations are {130!, 287298761874053529600, 38016, 64, 7, 2}, so a(130) = 64.
For n = 563, the iterations are {563!, 2875041108020454013464609906430286933482949481627276804096000000000, 77051520, 512, 10, 4, 3, 2}, so a(563) = 512.
		

Crossrefs

Cf. A000005, A000142, A000010, A051953, A036450-A036459, A380802, A380803.

Programs

Mathematica

Join[{1,2},Table[SelectFirst[Rest[NestWhileList[DivisorSigma[0,#]&,n!,#>2&]],IntegerQ[Log[2,#]]&],{n,3,100}]] (* Harvey P. Dale, Jul 02 2018 *)

PARI

a(n) = {my(m = n!); while(1 << valuation(m, 2) != m, m = numdiv(m)); m;} \\ Amiram Eldar, Feb 04 2025

Formula

a(n) = 2^A380802(n). - Amiram Eldar, Feb 04 2025